# Definition:Generalized Ordered Space/Definition 1

## Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $\tau$ be a topology for $S$.

$\left({S, \preceq, \tau}\right)$ is a generalized ordered space if and only if:

$(1): \quad \left({S, \tau}\right)$ is a Hausdorff space
$(2): \quad$ there exists a basis for $\left({S, \tau}\right)$ whose elements are convex in $S$.